If-4x > 20 and `xinz^(+)` then x belongs to

Options

A. {-6, -7,-8, ........}
B. `phi`
C. {-4, -3, -2, -1}
D. {1, 2, 3, 4, ........}

Answer

To solve the inequality \(-4x > 20\) with the condition that \(x\) belongs to positive integers (\(x \in \mathbb{Z}^+\)), we can follow these steps:

### Step 1: Rewrite the Inequality
We start with the inequality:
\[ -4x > 20 \]

### Step 2: Divide by -4
To isolate \(x\), we divide both sides of the inequality by \(-4\). Remember that when we divide or multiply an inequality by a negative number, the direction of the inequality sign changes:
\[ x < \frac{20}{-4} \]

### Step 3: Simplify the Right Side
Now, we simplify the right side:
\[ x < -5 \]

### Step 4: Analyze the Solution
The solution \(x < -5\) indicates that \(x\) can take values less than \(-5\). However, we are given that \(x\) must be a positive integer (\(x \in \mathbb{Z}^+\)).

### Step 5: Conclusion
Since there are no positive integers that are less than \(-5\), we conclude that there are no valid solutions for \(x\) under the given conditions. Therefore, \(x\) does not belong to any positive integers.

### Final Answer
Thus, \(x\) belongs to the empty set, which can be denoted as:
\[ x \in \emptyset \]